A new metamodeling approach is proposed to characterize the output (response) random process of a dynamic system with random variables, excited by input random processes. The metamodel is then used to efficiently estimate the time-dependent reliability. The input random processes are decomposed using principal components, and a few simulations are used to estimate the distributions of the decomposition coefficients. A similar decomposition is performed on the output random process. A Kriging model is then built between the input and output decomposition coefficients and is used subsequently to quantify the output random process. The innovation of our approach is that the system input is not deterministic but random. We establish, therefore, a surrogate model between the input and output random processes. To achieve this goal, we use an integral expression of the total probability theorem to estimate the marginal distribution of the output decomposition coefficients. The integral is efficiently estimated using a Monte Carlo (MC) approach which simulates from a mixture of sampling distributions with equal mixing probabilities. The quantified output random process is finally used to estimate the time-dependent probability of failure. The proposed method is illustrated with a corroding beam example.